Answer to Question #212975 in Electricity and Magnetism for Anita

Question #212975

A potential field is given by V = 3x^2y - yz. Which of the following is not true?

(a) At point (1, 0, -1) V and E vanish.

(b) x^2y = 1 is an equipotential line on the xy-plane.

(c) The equipotential surface V = - 8 passes through point P + 2, -1, 4)

(d) The electric field at P is 12ax - 8ay - az V/m.

(e) A unit normal to the equipotential surface V = - 8 at P is -0,83ax + 0,55ay+ 0,07az.


1
Expert's answer
2021-07-05T16:09:57-0400

Gives

"V=3x^2y-yz"

"V_{(1,0,1)}=0"

"E=-\\nabla V"


"E_{1,0,1}=-(6(0)a_x+(3\\times1+1)a_y-0a_z)"

"E=-4a_y"

Option (a) wrong(incorrect)

"P=x^2y-1"

Equpotential line equation


"\\nabla.P=\\nabla.(x^2y-1)=2xya_x+x^2a_y\\rightarrow(2)"


Equation (2) is lies xy-plane

Option (b) is correct option


Electric field

"V=3x^2y-yz"


"V_{(2,-1,4)}=3\\times4(-1)-(-1)\\times4=-8V"

Option (c) is correct


"E=-\\nabla V=-(6xya_x+(3x^2-z)a_y-ya_z)"

"E=12a_x-8a_y-a_z"

Option (d) correct

"E=12ax-8ay-az"

"\\hat{V}=\\frac{\\nabla V}{ |\\nabla V|}"

"\\hat{V}=\\frac{6axy a_x+(3x^2-z)a_y+a_z(-y)}{14.5}"


"\\hat{V}_{(2,-1,4)}=\\frac{-12}{14.4}a_x+\\frac{8}{14.4}a_y+\\frac{1}{14.4}a_z"

"\\hat{V}=-0.83a_x+0.55a_y+0.07a_z"

Option (e) is correct


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